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A Spatial-Temporal Model for the Evolution of the COVID-19 Pandemic in Spain Including Mobility

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A Spatial-Temporal Model for the Evolution of the COVID-19 Pandemic in Spain Including Mobility mathematics Article A Spatial-Temporal Model for the Evolution of the COVID-19 Pandemic in Spain Including Mobility Francesc Aràndiga, Antonio Baeza * , Isabel Cordero-Carrión , Rosa Donat, M. Carmen Martí, Pep Mulet and Dionisio F. Yáñez Departament de Matemàtiques, Universitat de València, Av. Vicent Andrés Estellés, E-46100 Burjassot, Spain; [email protected] (F.A.); [email protected] (I.C.-C.); [email protected] (R.D.); [email protected] (M.C.M.); [email protected] (P.M.); [email protected] (D.F.Y.) * Correspondence: [email protected] Received: 20 August 2020; Accepted: 23 September 2020; Published: 1 October 2020 Abstract: In this work, a model for the simulation of infectious disease outbreaks including mobility data is presented. The model is based on the SAIR compartmental model and includes mobility data terms that model the flow of people between different regions. The aim of the model is to analyze the influence of mobility on the evolution of a disease after a lockdown period and to study the appearance of small epidemic outbreaks due to the so-called imported cases. We apply the model to the simulation of the COVID-19 in the various areas of Spain, for which the authorities made available mobility data based on the position of cell phones. We also introduce a method for the estimation of incomplete mobility data. Some numerical experiments show the importance of data completion and indicate that the model is able to qualitatively simulate the spread tendencies of small outbreaks. This work was motivated by an open call made to the mathematical community in Spain to help predict the spread of the epidemic. Keywords: spatial-temporal SAIR model; mobility; outbreak spread; COVID-19; epidemic model; lockdown 1. Introduction 1.1. Motivation And Scope On 31 December 2019, the World Health Organization (WHO) office in the People’s Republic of China got an statement from the Wuhan Municipal Health Commission reporting cases of ‘viral pneumonia’ in Wuhan. Since then, the disease, known today as COVID-19, caused by the SARS-CoV-2 virus, has become a part of everyone’s daily life by now. The COVID-19 was declared pandemic by the WHO on 11 March 2020. In Spain, the first case of COVID-19 was diagnosed on 31 January 2020. During February, the number of infected people increased quickly, as well as the number of deaths, and on 14 March 2020 the state of alarm was declared and a lockdown was imposed on all the country as a drastic measure to dramatically reduce the spread of the disease. From that day on, people should stay at home and only indispensable trips within their hometowns were allowed. All flights from or to Spain were canceled, except the ones launched by the government bringing people home. However, some essential activities were allowed as, for instance road transportation of goods or harvesting fruits and vegetables to export them. These strict mobility measures were lessened gradually. A four-phase de-escalation plan was designed by the government and applied from 2 May 2020 to 22 June 2020, when the state of alarm ended, along with the lockdown and mobility restrictions. Each phase consisted on different contention Mathematics 2020, 8, 1677; doi:10.3390/math8101677 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 1677 2 of 19 policies. Regarding mobility, the most important changes came with the second phase (Phase 1) that allowed people to travel freely inside their province and with the third phase (Phase 2) that allowed people to travel freely through all the country. It is important to remark that mobility was permitted as long as no symptoms of the infection were shown. Answering the call from the Comité Español de Matemáticas, (CEMat, Spanish Committee of Mathematics, see http://matematicas.uclm.es/cemat/covid19/en/), it is the purpose of this paper to model the spreading of the COVID-19 virus in Spain, focusing on the impact of the mobility restrictions imposed to the population during the state of alarm. This work is based on previous research on the spread of the 2009 A/H1N1 influenza pandemic in Chile [1]. To estimate the impact of people’s mobility on the evolution and spread of the COVID-19 disease, it is really important to have access to real mobility data, so that models that include mobility data terms with parameters that can be tuned or estimated from said real data can be used. The Spanish National Institute of Statistics (INE) gave access to the research community to some mobility data obtained from the position of cell phones. These data were provided by the three major phone companies in Spain, covering around 80% of the total operating smartphones in the country. The provided data divide the country in smaller areas, depending on the density of population, and give information about the flux of people moving from their residency area to another one during the state of alarm. These data can be compared with the same fluxes happening on a normal day, before the mobility restrictions, to study how these mobility restrictions imposed really affected the spread of the infection. In accordance with the mobility data at hand, the model presented in this work divides the spatial domain under consideration into an arbitrary number C 2 N of smaller pieces, called mobility areas, corresponding to the mobility areas defined in the data provided by the INE or to aggregates of them. Moreover, the population will be also divided into groups by means of a compartmental model, that will be defined for each mobility area under consideration. The mobility model described in this work is strongly influenced by the nature of the mobility data supplied by the authorities. The model can be applied to any other country or region as far as similar mobility fluxes are available or can be reliably estimated. 1.2. Related Work The most used models for disease transmission are compartmental models. In compartmental models for epidemiology [2–4] a population is divided into groups (called compartments) according to the state in which individuals are regarding the disease. The model is defined by the type of compartments considered and by the statement of the ways in which individuals may change from one compartment to another. Both issues are strongly influenced by the particular disease under consideration, but different models, based on different hypotheses, can be raised to model the same outbreak. Qualitatively, there exist two major groups of compartmental models: deterministic and stochastic. Stochastic models [5] are based on the fact that some variables or parameters are random variables and the phenomenon being modelled is a stochastic process. Typical techniques used in these models are continuous-time Markov chains and stochastic differential equations. Deterministic models assume that the variables and parameters are continuous functions or constants, and the models are based on ordinary or partial differential equations. In this work we will focus on deterministic models. The state variables are the number of individuals in each group. One of the simplest model is the SIR model, first introduced in Reference [6]. This model consists on considering three compartments: • Susceptible (S): individuals that can be infected with the disease, • Infectious (I): individuals that have the disease and can transmit it, • Retired (R): individuals that have recovered from the disease, or have died. Mathematics 2020, 8, 1677 3 of 19 Note that this model considers either that the disease does not have an incubation period, or that individuals can transmit the disease during this period. No change in population because of new people born is included in the model. Individuals change compartment according to some fluxes (units are individuals per unit of time), typically governed by flux rates (percentage of individuals in one group that move to another group per unit of time). Flux rates are normally taken as constants to be determined from empirical data, through some kind of parameter estimation, but in a more realistic scenario non-constant flux rates have to be considered. If a perfect mixing of individuals in each compartment is assumed, the model can be posed as a system of Ordinary Differential Equations (ODE), where the left hand sides are the variations in each group and the right hand sides are built from fluxes. For instance, the SIR model corresponds to the ODE system IS S0 = −b , N IS I0 = b − gI, (1) N R0 = gI, with total population N = S + I + R and b, g positive constants. Important quantities in the model I are the infection rate b N (probability of getting infected in one unit of time) at which susceptible I individuals get infected, which is proportional to the proportion of infected individuals N through the contact rate b. The recovery rate g is equal to 1/TR, where TR is the mean infectious period measured in days. An interesting parameter of the model is the basic reproduction number, denoted by R0, and given b by R0 = g . At a very high level the behavior of an epidemic modeled by the SIR model is controlled by this parameter: if R0 > 1 then the number of (instant) infectious takes its maximum a certain time after the appearance of the epidemic, and then decreases, while if R0 < 1 then the infectious is always decreasing. In both cases the size of I tends to zero as time goes to infinity. A natural extension of the SIR model is the SEIR model [7], that includes Exposed (E compartment) individuals. This group models the incubation period of the disease, as exposed individuals have been infected but cannot transmit the disease to susceptible individuals.
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